# 先加载两个基本数据集
library(MASS)
library(ISLR2)

Attaching package: ‘ISLR2’


The following object is masked from ‘package:MASS’:

    Boston

# ISLR2包含Boston数据集，记录了波士顿506个人口普查区的房价中位数（medv），我们将使用12个预测指标来预测medv，如rm（每栋房子的平均房间数）、age（1940年之前建造的自住单元的比例）和lstat（社会经济地位较低的家庭的百分比）

head(Boston)

lm.fit <- lm(medv ~ lstat, data = Boston)

# 上面是正常写法，下面是使用attach函数的写法
# attach(Boston) 将Boston数据集中的变量放入到搜索路径中
# lm.fit <- lm(medv ~ lstat)

lm.fit

Call:
lm(formula = medv ~ lstat, data = Boston)

Coefficients:
(Intercept)        lstat  
      34.55        -0.95

summary(lm.fit)

Call:
lm(formula = medv ~ lstat, data = Boston)

Residuals:
    Min      1Q  Median      3Q     Max 
-15.168  -3.990  -1.318   2.034  24.500 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 34.55384    0.56263   61.41   <2e-16 ***
lstat       -0.95005    0.03873  -24.53   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.216 on 504 degrees of freedom
Multiple R-squared:  0.5441,	Adjusted R-squared:  0.5432 
F-statistic: 601.6 on 1 and 504 DF,  p-value: < 2.2e-16

# 访问lm对象的各个部分
names(x = lm.fit) #拟合模型的各个部分
coef(lm.fit) #所估计的系数
confint(lm.fit) #所估计的系数的置信区间

predict(lm.fit, data.frame(lstat = (c(5, 10, 15))),interval = "confidence") #预测值和置信区间
predict(lm.fit, data.frame(lstat = (c(5, 10, 15))),interval = "prediction") #预测值和预测区间

# 使用plot（）和abline（）函数绘制medv和lstat以及最小二乘回归线
plot(Boston$lstat, Boston$medv)
# 或者attach(Boston)之后直接使用plot(lstat, medv)
abline(lm.fit)

# lstat与medv之间存在一定的非线性关系

plot(Boston$lstat, Boston$medv)
abline(lm.fit, lwd = 3)

plot(Boston$lstat, Boston$medv)
abline(lm.fit, lwd = 3, col = "red") #

attach(Boston)
plot(lstat, medv, col = "red")
plot(lstat, medv, pch = 20)
plot(lstat, medv, pch = "+")
plot(1:20, 1:20, pch = 1:20) #pch参数可以设置点的形状

par(mfrow = c(2, 2))
plot(lm.fit)

predict(lm.fit)

residuals(lm.fit)

rstudent(lm.fit)

plot(predict(lm.fit), residuals(lm.fit))
plot(predict(lm.fit), rstudent(lm.fit))

plot(hatvalues(lm.fit))
which.max(hatvalues(lm.fit))

lm.fit <- lm(medv ~ lstat + age, data = Boston)
summary(lm.fit)

Call:
lm(formula = medv ~ lstat + age, data = Boston)

Residuals:
    Min      1Q  Median      3Q     Max 
-15.981  -3.978  -1.283   1.968  23.158 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 33.22276    0.73085  45.458  < 2e-16 ***
lstat       -1.03207    0.04819 -21.416  < 2e-16 ***
age          0.03454    0.01223   2.826  0.00491 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.173 on 503 degrees of freedom
Multiple R-squared:  0.5513,	Adjusted R-squared:  0.5495 
F-statistic:   309 on 2 and 503 DF,  p-value: < 2.2e-16

# 对所有的12个变量进行的多元线性回归

lm.fit <- lm(medv ~ ., data = Boston)
summary(lm.fit)

Call:
lm(formula = medv ~ ., data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.1304  -2.7673  -0.5814   1.9414  26.2526 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  41.617270   4.936039   8.431 3.79e-16 ***
crim         -0.121389   0.033000  -3.678 0.000261 ***
zn            0.046963   0.013879   3.384 0.000772 ***
indus         0.013468   0.062145   0.217 0.828520    
chas          2.839993   0.870007   3.264 0.001173 ** 
nox         -18.758022   3.851355  -4.870 1.50e-06 ***
rm            3.658119   0.420246   8.705  < 2e-16 ***
age           0.003611   0.013329   0.271 0.786595    
dis          -1.490754   0.201623  -7.394 6.17e-13 ***
rad           0.289405   0.066908   4.325 1.84e-05 ***
tax          -0.012682   0.003801  -3.337 0.000912 ***
ptratio      -0.937533   0.132206  -7.091 4.63e-12 ***
lstat        -0.552019   0.050659 -10.897  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.798 on 493 degrees of freedom
Multiple R-squared:  0.7343,	Adjusted R-squared:  0.7278 
F-statistic: 113.5 on 12 and 493 DF,  p-value: < 2.2e-16

?summary.lm  # 查看summary对象中的参数

summary.lm                package:stats                R Documentation

_S_u_m_m_a_r_i_z_i_n_g _L_i_n_e_a_r _M_o_d_e_l _F_i_t_s

_D_e_s_c_r_i_p_t_i_o_n:

     ‘summary’ method for class ‘"lm"’.

_U_s_a_g_e:

     ## S3 method for class 'lm'
     summary(object, correlation = FALSE, symbolic.cor = FALSE, ...)
     
     ## S3 method for class 'summary.lm'
     print(x, digits = max(3, getOption("digits") - 3),
           symbolic.cor = x$symbolic.cor,
           signif.stars = getOption("show.signif.stars"), ...)
     
_A_r_g_u_m_e_n_t_s:

  object: an object of class ‘"lm"’, usually, a result of a call to
          ‘lm’.

       x: an object of class ‘"summary.lm"’, usually, a result of a
          call to ‘summary.lm’.

correlation: logical; if ‘TRUE’, the correlation matrix of the
          estimated parameters is returned and printed.

  digits: the number of significant digits to use when printing.

symbolic.cor: logical. If ‘TRUE’, print the correlations in a symbolic
          form (see ‘symnum’) rather than as numbers.

signif.stars: logical. If ‘TRUE’, ‘significance stars’ are printed for
          each coefficient.

     ...: further arguments passed to or from other methods.

_D_e_t_a_i_l_s:

     ‘print.summary.lm’ tries to be smart about formatting the
     coefficients, standard errors, etc. and additionally gives
     ‘significance stars’ if ‘signif.stars’ is ‘TRUE’.

     Aliased coefficients are omitted in the returned object but
     restored by the ‘print’ method.

     Correlations are printed to two decimal places (or symbolically):
     to see the actual correlations print ‘summary(object)$correlation’
     directly.

_V_a_l_u_e:

     The function ‘summary.lm’ computes and returns a list of summary
     statistics of the fitted linear model given in ‘object’, using the
     components (list elements) ‘"call"’ and ‘"terms"’ from its
     argument, plus

residuals: the _weighted_ residuals, the usual residuals rescaled by
          the square root of the weights specified in the call to ‘lm’.

coefficients: a p x 4 matrix with columns for the estimated
          coefficient, its standard error, t-statistic and
          corresponding (two-sided) p-value.  Aliased coefficients are
          omitted.

 aliased: named logical vector showing if the original coefficients are
          aliased.

   sigma: the square root of the estimated variance of the random error

                       sigma^2 = 1/(n-p) Sum(w[i] R[i]^2),              
          
          where R[i] is the i-th residual, ‘residuals[i]’.

      df: degrees of freedom, a 3-vector (p, n-p, p*), the first being
          the number of non-aliased coefficients, the last being the
          total number of coefficients.

fstatistic: (for models including non-intercept terms) a 3-vector with
          the value of the F-statistic with its numerator and
          denominator degrees of freedom.

r.squared: R^2, the ‘fraction of variance explained by the model’,

                    R^2 = 1 - Sum(R[i]^2) / Sum((y[i]- y*)^2),          
          
          where y* is the mean of y[i] if there is an intercept and
          zero otherwise.

adj.r.squared: the above R^2 statistic ‘_adjusted_’, penalizing for
          higher p.

cov.unscaled: a p x p matrix of (unscaled) covariances of the coef[j],
          j=1, ..., p.

correlation: the correlation matrix corresponding to the above
          ‘cov.unscaled’, if ‘correlation = TRUE’ is specified.

symbolic.cor: (only if ‘correlation’ is true.)  The value of the
          argument ‘symbolic.cor’.

na.action: from ‘object’, if present there.

_S_e_e _A_l_s_o:

     The model fitting function ‘lm’, ‘summary’.

     Function ‘coef’ will extract the matrix of coefficients with
     standard errors, t-statistics and p-values.

_E_x_a_m_p_l_e_s:

     ##-- Continuing the  lm(.) example:
     coef(lm.D90)  # the bare coefficients
     sld90 <- summary(lm.D90 <- lm(weight ~ group -1))  # omitting intercept
     sld90
     coef(sld90)  # much more
     
     ## model with *aliased* coefficient:
     lm.D9. <- lm(weight ~ group + I(group != "Ctl"))
     Sm.D9. <- summary(lm.D9.)
     Sm.D9. #  shows the NA NA NA NA  line
     stopifnot(length(cc <- coef(lm.D9.)) == 3, is.na(cc[3]),
               dim(coef(Sm.D9.)) == c(2,4), Sm.D9.$df == c(2, 18, 3))

summary(object = lm.fit)

Call:
lm(formula = medv ~ ., data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.1304  -2.7673  -0.5814   1.9414  26.2526 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  41.617270   4.936039   8.431 3.79e-16 ***
crim         -0.121389   0.033000  -3.678 0.000261 ***
zn            0.046963   0.013879   3.384 0.000772 ***
indus         0.013468   0.062145   0.217 0.828520    
chas          2.839993   0.870007   3.264 0.001173 ** 
nox         -18.758022   3.851355  -4.870 1.50e-06 ***
rm            3.658119   0.420246   8.705  < 2e-16 ***
age           0.003611   0.013329   0.271 0.786595    
dis          -1.490754   0.201623  -7.394 6.17e-13 ***
rad           0.289405   0.066908   4.325 1.84e-05 ***
tax          -0.012682   0.003801  -3.337 0.000912 ***
ptratio      -0.937533   0.132206  -7.091 4.63e-12 ***
lstat        -0.552019   0.050659 -10.897  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.798 on 493 degrees of freedom
Multiple R-squared:  0.7343,	Adjusted R-squared:  0.7278 
F-statistic: 113.5 on 12 and 493 DF,  p-value: < 2.2e-16

summary(lm.fit)$r.sq # 模型的R方，未矫正前的 
summary(lm.fit)$sigma # 模型的RSE，残差标准误差

# car包中的vif函数，可以用于计算方差膨胀因子，variance infation factors
# install.packages("car")
library(car)
vif(lm.fit)

# 使用vif函数取不同自变量的方差膨胀因子，并进行开根号。如果大于2，代表自变量与其他变量存在共线性，此时需要对模型进行优化

Updating HTML index of packages in '.Library'

Making 'packages.html' ...
 done

Loading required package: carData

print(vif(lm.fit))

    crim       zn    indus     chas      nox       rm      age      dis 
1.767486 2.298459 3.987181 1.071168 4.369093 1.912532 3.088232 3.954037 
     rad      tax  ptratio    lstat 
7.445301 9.002158 1.797060 2.870777

lm.fit1 <- lm(medv ~ . - age, data = Boston)
summary(lm.fit1)

Call:
lm(formula = medv ~ . - age, data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.1851  -2.7330  -0.6116   1.8555  26.3838 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  41.525128   4.919684   8.441 3.52e-16 ***
crim         -0.121426   0.032969  -3.683 0.000256 ***
zn            0.046512   0.013766   3.379 0.000785 ***
indus         0.013451   0.062086   0.217 0.828577    
chas          2.852773   0.867912   3.287 0.001085 ** 
nox         -18.485070   3.713714  -4.978 8.91e-07 ***
rm            3.681070   0.411230   8.951  < 2e-16 ***
dis          -1.506777   0.192570  -7.825 3.12e-14 ***
rad           0.287940   0.066627   4.322 1.87e-05 ***
tax          -0.012653   0.003796  -3.333 0.000923 ***
ptratio      -0.934649   0.131653  -7.099 4.39e-12 ***
lstat        -0.547409   0.047669 -11.483  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.794 on 494 degrees of freedom
Multiple R-squared:  0.7343,	Adjusted R-squared:  0.7284 
F-statistic: 124.1 on 11 and 494 DF,  p-value: < 2.2e-16

lm.fit2 <- lm(medv ~ . - age - indus, data = Boston)
summary(lm.fit2)

Call:
lm(formula = medv ~ . - age - indus, data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.1814  -2.7625  -0.6243   1.8448  26.3920 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  41.451747   4.903283   8.454 3.18e-16 ***
crim         -0.121665   0.032919  -3.696 0.000244 ***
zn            0.046191   0.013673   3.378 0.000787 ***
chas          2.871873   0.862591   3.329 0.000935 ***
nox         -18.262427   3.565247  -5.122 4.33e-07 ***
rm            3.672957   0.409127   8.978  < 2e-16 ***
dis          -1.515951   0.187675  -8.078 5.08e-15 ***
rad           0.283932   0.063945   4.440 1.11e-05 ***
tax          -0.012292   0.003407  -3.608 0.000340 ***
ptratio      -0.930961   0.130423  -7.138 3.39e-12 ***
lstat        -0.546509   0.047442 -11.519  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.789 on 495 degrees of freedom
Multiple R-squared:  0.7342,	Adjusted R-squared:  0.7289 
F-statistic: 136.8 on 10 and 495 DF,  p-value: < 2.2e-16

# 或者，也可以使用update函数 # update and (by default) re-fit a model
lm.fit3 <- update(lm.fit, ~ . - age)
summary(lm.fit3)

Call:
lm(formula = medv ~ crim + zn + indus + chas + nox + rm + dis + 
    rad + tax + ptratio + lstat, data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.1851  -2.7330  -0.6116   1.8555  26.3838 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  41.525128   4.919684   8.441 3.52e-16 ***
crim         -0.121426   0.032969  -3.683 0.000256 ***
zn            0.046512   0.013766   3.379 0.000785 ***
indus         0.013451   0.062086   0.217 0.828577    
chas          2.852773   0.867912   3.287 0.001085 ** 
nox         -18.485070   3.713714  -4.978 8.91e-07 ***
rm            3.681070   0.411230   8.951  < 2e-16 ***
dis          -1.506777   0.192570  -7.825 3.12e-14 ***
rad           0.287940   0.066627   4.322 1.87e-05 ***
tax          -0.012653   0.003796  -3.333 0.000923 ***
ptratio      -0.934649   0.131653  -7.099 4.39e-12 ***
lstat        -0.547409   0.047669 -11.483  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.794 on 494 degrees of freedom
Multiple R-squared:  0.7343,	Adjusted R-squared:  0.7284 
F-statistic: 124.1 on 11 and 494 DF,  p-value: < 2.2e-16

summary(lm(medv ~ lstat * age, data = Boston))

Call:
lm(formula = medv ~ lstat * age, data = Boston)

Residuals:
    Min      1Q  Median      3Q     Max 
-15.806  -4.045  -1.333   2.085  27.552 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) 36.0885359  1.4698355  24.553  < 2e-16 ***
lstat       -1.3921168  0.1674555  -8.313 8.78e-16 ***
age         -0.0007209  0.0198792  -0.036   0.9711    
lstat:age    0.0041560  0.0018518   2.244   0.0252 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.149 on 502 degrees of freedom
Multiple R-squared:  0.5557,	Adjusted R-squared:  0.5531 
F-statistic: 209.3 on 3 and 502 DF,  p-value: < 2.2e-16

lm.fit2 <- lm(medv ~ lstat + I(lstat^2),data = Boston) # 不能写成medv ~ lstat + lstat^2
summary(lm.fit2)

Call:
lm(formula = medv ~ lstat + I(lstat^2), data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-15.2834  -3.8313  -0.5295   2.3095  25.4148 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 42.862007   0.872084   49.15   <2e-16 ***
lstat       -2.332821   0.123803  -18.84   <2e-16 ***
I(lstat^2)   0.043547   0.003745   11.63   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.524 on 503 degrees of freedom
Multiple R-squared:  0.6407,	Adjusted R-squared:  0.6393 
F-statistic: 448.5 on 2 and 503 DF,  p-value: < 2.2e-16

lm.fit <- lm(medv ~ lstat,data = Boston)
summary(lm.fit)
anova(lm.fit, lm.fit2)

Call:
lm(formula = medv ~ lstat, data = Boston)

Residuals:
    Min      1Q  Median      3Q     Max 
-15.168  -3.990  -1.318   2.034  24.500 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 34.55384    0.56263   61.41   <2e-16 ***
lstat       -0.95005    0.03873  -24.53   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.216 on 504 degrees of freedom
Multiple R-squared:  0.5441,	Adjusted R-squared:  0.5432 
F-statistic: 601.6 on 1 and 504 DF,  p-value: < 2.2e-16

lm.fit <- lm(medv ~ lstat,data = Boston)
lm.fit2 <- lm(medv ~ lstat + I(lstat^2),data = Boston)
anova(lm.fit, lm.fit2)

par(mfrow = c(2, 2))
plot(lm.fit2)

lm.fit5 <- lm(medv ~ poly(lstat, 5),data = Boston)
summary(lm.fit5)

Call:
lm(formula = medv ~ poly(lstat, 5), data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-13.5433  -3.1039  -0.7052   2.0844  27.1153 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)       22.5328     0.2318  97.197  < 2e-16 ***
poly(lstat, 5)1 -152.4595     5.2148 -29.236  < 2e-16 ***
poly(lstat, 5)2   64.2272     5.2148  12.316  < 2e-16 ***
poly(lstat, 5)3  -27.0511     5.2148  -5.187 3.10e-07 ***
poly(lstat, 5)4   25.4517     5.2148   4.881 1.42e-06 ***
poly(lstat, 5)5  -19.2524     5.2148  -3.692 0.000247 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.215 on 500 degrees of freedom
Multiple R-squared:  0.6817,	Adjusted R-squared:  0.6785 
F-statistic: 214.2 on 5 and 500 DF,  p-value: < 2.2e-16

# 当然，我们并不局限于使用多项式变换来进行预测。这里我们尝试一个对数变换
summary(lm(medv ~ log(lstat), data = Boston))
summary(lm(medv ~ log(rm), data = Boston))

Call:
lm(formula = medv ~ log(lstat), data = Boston)

Residuals:
     Min       1Q   Median       3Q      Max 
-14.4599  -3.5006  -0.6686   2.1688  26.0129 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  52.1248     0.9652   54.00   <2e-16 ***
log(lstat)  -12.4810     0.3946  -31.63   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.329 on 504 degrees of freedom
Multiple R-squared:  0.6649,	Adjusted R-squared:  0.6643 
F-statistic:  1000 on 1 and 504 DF,  p-value: < 2.2e-16

Call:
lm(formula = medv ~ log(rm), data = Boston)

Residuals:
    Min      1Q  Median      3Q     Max 
-19.487  -2.875  -0.104   2.837  39.816 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -76.488      5.028  -15.21   <2e-16 ***
log(rm)       54.055      2.739   19.73   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.915 on 504 degrees of freedom
Multiple R-squared:  0.4358,	Adjusted R-squared:  0.4347 
F-statistic: 389.3 on 1 and 504 DF,  p-value: < 2.2e-16

# 生成正交多项式特征
x <- 1:10
poly_features <- poly(x, degree = 3)
print(poly_features)

# 生成原始多项式特征
raw_poly_features <- poly(x, degree = 3, raw = TRUE)
print(raw_poly_features)

                1           2          3
 [1,] -0.49543369  0.52223297 -0.4534252
 [2,] -0.38533732  0.17407766  0.1511417
 [3,] -0.27524094 -0.08703883  0.3778543
 [4,] -0.16514456 -0.26111648  0.3346710
 [5,] -0.05504819 -0.34815531  0.1295501
 [6,]  0.05504819 -0.34815531 -0.1295501
 [7,]  0.16514456 -0.26111648 -0.3346710
 [8,]  0.27524094 -0.08703883 -0.3778543
 [9,]  0.38533732  0.17407766 -0.1511417
[10,]  0.49543369  0.52223297  0.4534252
attr(,"coefs")
attr(,"coefs")$alpha
[1] 5.5 5.5 5.5

attr(,"coefs")$norm2
[1]    1.0   10.0   82.5  528.0 3088.8

attr(,"degree")
[1] 1 2 3
attr(,"class")
[1] "poly"   "matrix"
       1   2    3
 [1,]  1   1    1
 [2,]  2   4    8
 [3,]  3   9   27
 [4,]  4  16   64
 [5,]  5  25  125
 [6,]  6  36  216
 [7,]  7  49  343
 [8,]  8  64  512
 [9,]  9  81  729
[10,] 10 100 1000
attr(,"degree")
[1] 1 2 3
attr(,"class")
[1] "poly"   "matrix"

# 现在使用的是其他的数据，ISLR2库中的Carseats数据集 
# 我们将尝试基于一些预测因素来预测400个地点的销售Sales（儿童汽车座椅的销售）
head(Carseats)

lm.fit <- lm(Sales ~ . + Income:Advertising + Price:Age, data = Carseats) # Income:Advertising表示Income和Advertising的交互项，以及原始数据中的定性项：ShelveLoc	
summary(lm.fit)

Call:
lm(formula = Sales ~ . + Income:Advertising + Price:Age, data = Carseats)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.9208 -0.7503  0.0177  0.6754  3.3413 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)         6.5755654  1.0087470   6.519 2.22e-10 ***
CompPrice           0.0929371  0.0041183  22.567  < 2e-16 ***
Income              0.0108940  0.0026044   4.183 3.57e-05 ***
Advertising         0.0702462  0.0226091   3.107 0.002030 ** 
Population          0.0001592  0.0003679   0.433 0.665330    
Price              -0.1008064  0.0074399 -13.549  < 2e-16 ***
ShelveLocGood       4.8486762  0.1528378  31.724  < 2e-16 ***
ShelveLocMedium     1.9532620  0.1257682  15.531  < 2e-16 ***
Age                -0.0579466  0.0159506  -3.633 0.000318 ***
Education          -0.0208525  0.0196131  -1.063 0.288361    
UrbanYes            0.1401597  0.1124019   1.247 0.213171    
USYes              -0.1575571  0.1489234  -1.058 0.290729    
Income:Advertising  0.0007510  0.0002784   2.698 0.007290 ** 
Price:Age           0.0001068  0.0001333   0.801 0.423812    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.011 on 386 degrees of freedom
Multiple R-squared:  0.8761,	Adjusted R-squared:  0.8719 
F-statistic:   210 on 13 and 386 DF,  p-value: < 2.2e-16

attach(Carseats)
contrasts(ShelveLoc)

LoadLibraries <- function(){
    library(ISLR2)
    library(MASS)
    print("the libraries ISLR2 & MASS have been loaded")
}

LoadLibraries  #该函数的信息
LoadLibraries() #该函数的执行

function () 
{
    library(ISLR2)
    library(MASS)
    print("the libraries ISLR2 & MASS have been loaded")
}

[1] "the libraries ISLR2 & MASS have been loaded"

?outer

outer                   package:base                   R Documentation

_O_u_t_e_r _P_r_o_d_u_c_t _o_f _A_r_r_a_y_s

_D_e_s_c_r_i_p_t_i_o_n:

     The outer product of the arrays ‘X’ and ‘Y’ is the array ‘A’ with
     dimension ‘c(dim(X), dim(Y))’ where element ‘A[c(arrayindex.x,
     arrayindex.y)] = FUN(X[arrayindex.x], Y[arrayindex.y], ...)’.

_U_s_a_g_e:

     outer(X, Y, FUN = "*", ...)
     X %o% Y
     
_A_r_g_u_m_e_n_t_s:

    X, Y: first and second arguments for function ‘FUN’.  Typically a
          vector or array.

     FUN: a function to use on the outer products, found _via_
          ‘match.fun’ (except for the special case ‘"*"’).

     ...: optional arguments to be passed to ‘FUN’.

_D_e_t_a_i_l_s:

     ‘X’ and ‘Y’ must be suitable arguments for ‘FUN’.  Each will be
     extended by ‘rep’ to length the products of the lengths of ‘X’ and
     ‘Y’ before ‘FUN’ is called.

     ‘FUN’ is called with these two extended vectors as arguments (plus
     any arguments in ‘...’).  It must be a vectorized function (or the
     name of one) expecting at least two arguments and returning a
     value with the same length as the first (and the second).

     Where they exist, the [dim]names of ‘X’ and ‘Y’ will be copied to
     the answer, and a dimension assigned which is the concatenation of
     the dimensions of ‘X’ and ‘Y’ (or lengths if dimensions do not
     exist).

     ‘FUN = "*"’ is handled as a special case _via_ ‘as.vector(X) %*%
     t(as.vector(Y))’, and is intended only for numeric vectors and
     arrays.

     ‘%o%’ is binary operator providing a wrapper for ‘outer(x, y,
     "*")’.

_A_u_t_h_o_r(_s):

     Jonathan Rougier

_R_e_f_e_r_e_n_c_e_s:

     Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S
     Language_.  Wadsworth & Brooks/Cole.

_S_e_e _A_l_s_o:

     ‘%*%’ for usual (_inner_) matrix vector multiplication;
     ‘kronecker’ which is based on ‘outer’; ‘Vectorize’ for vectorizing
     a non-vectorized function.

_E_x_a_m_p_l_e_s:

     x <- 1:9; names(x) <- x
     # Multiplication & Power Tables
     x %o% x
     y <- 2:8; names(y) <- paste(y,":", sep = "")
     outer(y, x, `^`)
     
     outer(month.abb, 1999:2003, FUN = paste)
     
     ## three way multiplication table:
     x %o% x %o% y[1:3]

# a
# 固定IQ和GPA的取值，然后考虑预测变量的取值对于响应变量的影响 
 
library(plotly)   #用于构建交互式的图表
salary_model <- function(GPA,IQ,LEVEL){
  50+20*GPA+0.07*IQ+35*LEVEL+0.01*GPA*IQ-10*GPA*LEVEL
}
gpa_test <- seq(1,5,length=10)  # 人工构建从1到5，长度为10的等间隔的IQ值
iq_test <- seq(1,200,length=20) # 人工构建从1到200，长度为20的等间隔的GPA值

print(gpa_test)
print(iq_test)

 [1] 1.000000 1.444444 1.888889 2.333333 2.777778 3.222222 3.666667 4.111111
 [9] 4.555556 5.000000
 [1]   1.00000  23.11111  45.22222  67.33333  89.44444 111.55556 133.66667
 [8] 155.77778 177.88889 200.00000

outer(gpa_test,iq_test,LEVEL=1,FUN=salary_model)   
t(outer(gpa_test,iq_test,LEVEL=1,FUN=salary_model))

?add_surface
?plot_ly

add_trace                package:plotly                R Documentation

_A_d_d _t_r_a_c_e(_s) _t_o _a _p_l_o_t_l_y _v_i_s_u_a_l_i_z_a_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     Add trace(s) to a plotly visualization

_U_s_a_g_e:

     add_trace(p, ..., data = NULL, inherit = TRUE)
     
     add_markers(p, x = NULL, y = NULL, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_text(
       p,
       x = NULL,
       y = NULL,
       z = NULL,
       text = NULL,
       ...,
       data = NULL,
       inherit = TRUE
     )
     
     add_paths(p, x = NULL, y = NULL, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_lines(p, x = NULL, y = NULL, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_segments(
       p,
       x = NULL,
       y = NULL,
       xend = NULL,
       yend = NULL,
       ...,
       data = NULL,
       inherit = TRUE
     )
     
     add_polygons(p, x = NULL, y = NULL, ..., data = NULL, inherit = TRUE)
     
     add_sf(p, ..., x = ~x, y = ~y, data = NULL, inherit = TRUE)
     
     add_table(p, ..., rownames = TRUE, data = NULL, inherit = TRUE)
     
     add_ribbons(
       p,
       x = NULL,
       ymin = NULL,
       ymax = NULL,
       ...,
       data = NULL,
       inherit = TRUE
     )
     
     add_image(p, z = NULL, colormodel = NULL, ..., data = NULL, inherit = TRUE)
     
     add_area(p, r = NULL, theta = NULL, t = NULL, ..., data = NULL, inherit = TRUE)
     
     add_pie(p, values = NULL, labels = NULL, ..., data = NULL, inherit = TRUE)
     
     add_bars(p, x = NULL, y = NULL, ..., data = NULL, inherit = TRUE)
     
     add_histogram(p, x = NULL, y = NULL, ..., data = NULL, inherit = TRUE)
     
     add_histogram2d(
       p,
       x = NULL,
       y = NULL,
       z = NULL,
       ...,
       data = NULL,
       inherit = TRUE
     )
     
     add_histogram2dcontour(
       p,
       x = NULL,
       y = NULL,
       z = NULL,
       ...,
       data = NULL,
       inherit = TRUE
     )
     
     add_heatmap(p, x = NULL, y = NULL, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_contour(p, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_boxplot(p, x = NULL, y = NULL, ..., data = NULL, inherit = TRUE)
     
     add_surface(p, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_mesh(p, x = NULL, y = NULL, z = NULL, ..., data = NULL, inherit = TRUE)
     
     add_scattergeo(p, ...)
     
     add_choropleth(p, z = NULL, ..., data = NULL, inherit = TRUE)
     
_A_r_g_u_m_e_n_t_s:

       p: a plotly object

     ...: Arguments (i.e., attributes) passed along to the trace
          ‘type’. See ‘schema()’ for a list of acceptable attributes
          for a given trace ‘type’ (by going to ‘traces’ -> ‘type’ ->
          ‘attributes’). Note that attributes provided at this level
          may override other arguments (e.g. ‘plot_ly(x = 1:10, y =
          1:10, color = I("red"), marker = list(color = "blue"))’).

    data: A data frame (optional) or crosstalk::SharedData object.

 inherit: inherit attributes from ‘plot_ly()’?

       x: the x variable.

       y: the y variable.

       z: a numeric matrix (unless ‘add_image()’, which wants a raster
          object, see ‘as.raster()’).

    text: textual labels.

    xend: "final" x position (in this context, x represents "start")

    yend: "final" y position (in this context, y represents "start")

rownames: whether or not to display the rownames of ‘data’.

    ymin: a variable used to define the lower boundary of a polygon.

    ymax: a variable used to define the upper boundary of a polygon.

colormodel: Sets the colormodel for image traces if ‘z’ is not a raster
          object. If ‘z’ is a raster object (see ‘as.raster()’), the
          ‘'rgba'’ colormodel is always used.

       r: Sets the radial coordinates.

   theta: Sets the angular coordinates.

       t: Deprecated. Use ‘theta’ instead.

  values: the value to associated with each slice of the pie.

  labels: the labels (categories) corresponding to ‘values’.

_A_u_t_h_o_r(_s):

     Carson Sievert

_R_e_f_e_r_e_n_c_e_s:

     <https://plotly-r.com/overview.html>

     <https://plotly.com/r/>

     <https://plotly.com/r/reference/>

_S_e_e _A_l_s_o:

     ‘plot_ly()’

_E_x_a_m_p_l_e_s:

     # the `plot_ly()` function initiates an object, and if no trace type
     # is specified, it sets a sensible default
     p <- plot_ly(economics, x = ~date, y = ~uempmed)
     p
     
     # some `add_*()` functions are a specific case of a trace type
     # for example, `add_markers()` is a scatter trace with mode of markers
     add_markers(p)
     
     # scatter trace with mode of text
     add_text(p, text = "%")
     
     # scatter trace with mode of lines 
     add_paths(p)
     
     # like `add_paths()`, but ensures points are connected according to `x`
     add_lines(p)
     
     # if you prefer to work with plotly.js more directly, can always
     # use `add_trace()` and specify the type yourself
     add_trace(p, type = "scatter", mode = "markers+lines")
     
     # mappings provided to `plot_ly()` are "global", but can be overwritten
     plot_ly(economics, x = ~date, y = ~uempmed, color = I("red"), showlegend = FALSE) %>% 
       add_lines() %>%
       add_markers(color = ~pop)
     
     # a number of `add_*()` functions are special cases of the scatter trace
     plot_ly(economics, x = ~date) %>% 
       add_ribbons(ymin = ~pce - 1e3, ymax = ~pce + 1e3)
     
     # use `group_by()` (or `group2NA()`) to apply visual mapping
     # once per group (e.g. one line per group)
     txhousing %>% 
       group_by(city) %>% 
       plot_ly(x = ~date, y = ~median) %>%
       add_lines(color = I("black"))
     
     ## Not run:
     
     # use `add_sf()` or `add_polygons()` to create geo-spatial maps
     # http://blog.cpsievert.me/2018/03/30/visualizing-geo-spatial-data-with-sf-and-plotly/
     if (requireNamespace("sf", quietly = TRUE)) {
       nc <- sf::st_read(system.file("shape/nc.shp", package = "sf"), quiet = TRUE)
       plot_ly() %>% add_sf(data = nc)
     }
     
     # univariate summary statistics
     plot_ly(mtcars, x = ~factor(vs), y = ~mpg) %>% 
       add_boxplot()
     plot_ly(mtcars, x = ~factor(vs), y = ~mpg) %>% 
       add_trace(type = "violin")
       
     # `add_histogram()` does binning for you...
     mtcars %>%
       plot_ly(x = ~factor(vs)) %>%
       add_histogram()
       
     # ...but you can 'pre-compute' bar heights in R
     mtcars %>%
       dplyr::count(vs) %>%
       plot_ly(x = ~vs, y = ~n) %>%
       add_bars()
     
     # the 2d analogy of add_histogram() is add_histogram2d()/add_histogram2dcontour()
     library(MASS)
     (p <- plot_ly(geyser, x = ~waiting, y = ~duration))
     add_histogram2d(p)
     add_histogram2dcontour(p)
     
     # the 2d analogy of add_bars() is add_heatmap()/add_contour()
     # (i.e., bin counts must be pre-specified)
     den <- kde2d(geyser$waiting, geyser$duration)
     p <- plot_ly(x = den$x, y = den$y, z = den$z)
     add_heatmap(p)
     add_contour(p)
     
     # `add_table()` makes it easy to map a data frame to the table trace type
     plot_ly(economics) %>% 
       add_table()
     
     # pie charts!
     ds <- data.frame(labels = c("A", "B", "C"), values = c(10, 40, 60))
     plot_ly(ds, labels = ~labels, values = ~values) %>%
       add_pie() %>%
       layout(title = "Basic Pie Chart using Plotly")
       
     data(wind)
     plot_ly(wind, r = ~r, theta = ~t) %>% 
       add_area(color = ~nms) %>%
       layout(
         polar = list(
           radialaxis = list(ticksuffix = "%"), 
           angularaxis = list(rotation = 90)
         )
       )
     
     # ------------------------------------------------------------
     # 3D chart types
     # ------------------------------------------------------------
     plot_ly(z = ~volcano) %>% 
       add_surface()
     plot_ly(x = c(0, 0, 1), y = c(0, 1, 0), z = c(0, 0, 0)) %>% 
       add_mesh()
     ## End(Not run)

plot_ly                 package:plotly                 R Documentation

_I_n_i_t_i_a_t_e _a _p_l_o_t_l_y _v_i_s_u_a_l_i_z_a_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     This function maps R objects to plotly.js, an (MIT licensed)
     web-based interactive charting library. It provides abstractions
     for doing common things (e.g. mapping data values to fill colors
     (via ‘color’) or creating animations (via ‘frame’)) and sets some
     different defaults to make the interface feel more 'R-like' (i.e.,
     closer to ‘plot()’ and ‘ggplot2::qplot()’).

_U_s_a_g_e:

     plot_ly(
       data = data.frame(),
       ...,
       type = NULL,
       name,
       color,
       colors = NULL,
       alpha = NULL,
       stroke,
       strokes = NULL,
       alpha_stroke = 1,
       size,
       sizes = c(10, 100),
       span,
       spans = c(1, 20),
       symbol,
       symbols = NULL,
       linetype,
       linetypes = NULL,
       split,
       frame,
       width = NULL,
       height = NULL,
       source = "A"
     )
     
_A_r_g_u_m_e_n_t_s:

    data: A data frame (optional) or crosstalk::SharedData object.

     ...: Arguments (i.e., attributes) passed along to the trace
          ‘type’. See ‘schema()’ for a list of acceptable attributes
          for a given trace ‘type’ (by going to ‘traces’ -> ‘type’ ->
          ‘attributes’). Note that attributes provided at this level
          may override other arguments (e.g. ‘plot_ly(x = 1:10, y =
          1:10, color = I("red"), marker = list(color = "blue"))’).

    type: A character string specifying the trace type (e.g.
          ‘"scatter"’, ‘"bar"’, ‘"box"’, etc). If specified, it
          _always_ creates a trace, otherwise

    name: Values mapped to the trace's name attribute. Since a trace
          can only have one name, this argument acts very much like
          ‘split’ in that it creates one trace for every unique value.

   color: Values mapped to relevant 'fill-color' attribute(s) (e.g.
          fillcolor, marker.color, textfont.color, etc.). The mapping
          from data values to color codes may be controlled using
          ‘colors’ and ‘alpha’, or avoided altogether via ‘I()’ (e.g.,
          ‘color = I("red")’). Any color understood by
          ‘grDevices::col2rgb()’ may be used in this way.

  colors: Either a colorbrewer2.org palette name (e.g. "YlOrRd" or
          "Blues"), or a vector of colors to interpolate in hexadecimal
          "#RRGGBB" format, or a color interpolation function like
          ‘colorRamp()’.

   alpha: A number between 0 and 1 specifying the alpha channel applied
          to ‘color’. Defaults to 0.5 when mapping to fillcolor and 1
          otherwise.

  stroke: Similar to ‘color’, but values are mapped to relevant
          'stroke-color' attribute(s) (e.g., marker.line.color and
          line.color for filled polygons). If not specified, ‘stroke’
          inherits from ‘color’.

 strokes: Similar to ‘colors’, but controls the ‘stroke’ mapping.

alpha_stroke: Similar to ‘alpha’, but applied to ‘stroke’.

    size: (Numeric) values mapped to relevant 'fill-size' attribute(s)
          (e.g., marker.size, textfont.size, and error_x.width). The
          mapping from data values to symbols may be controlled using
          ‘sizes’, or avoided altogether via ‘I()’ (e.g., ‘size =
          I(30)’).

   sizes: A numeric vector of length 2 used to scale ‘size’ to pixels.

    span: (Numeric) values mapped to relevant 'stroke-size'
          attribute(s) (e.g., marker.line.width, line.width for filled
          polygons, and error_x.thickness) The mapping from data values
          to symbols may be controlled using ‘spans’, or avoided
          altogether via ‘I()’ (e.g., ‘span = I(30)’).

   spans: A numeric vector of length 2 used to scale ‘span’ to pixels.

  symbol: (Discrete) values mapped to marker.symbol. The mapping from
          data values to symbols may be controlled using ‘symbols’, or
          avoided altogether via ‘I()’ (e.g., ‘symbol =
          I("pentagon")’). Any pch value or symbol name may be used in
          this way.

 symbols: A character vector of pch values or symbol names.

linetype: (Discrete) values mapped to line.dash. The mapping from data
          values to symbols may be controlled using ‘linetypes’, or
          avoided altogether via ‘I()’ (e.g., ‘linetype = I("dash")’).
          Any ‘lty’ (see par) value or dash name may be used in this
          way.

linetypes: A character vector of ‘lty’ values or dash names

   split: (Discrete) values used to create multiple traces (one trace
          per value).

   frame: (Discrete) values used to create animation frames.

   width: Width in pixels (optional, defaults to automatic sizing).

  height: Height in pixels (optional, defaults to automatic sizing).

  source: a character string of length 1. Match the value of this
          string with the source argument in ‘event_data()’ to retrieve
          the event data corresponding to a specific plot (shiny apps
          can have multiple plots).

_D_e_t_a_i_l_s:

     Unless ‘type’ is specified, this function just initiates a plotly
     object with 'global' attributes that are passed onto downstream
     uses of ‘add_trace()’ (or similar). A formula must always be used
     when referencing column name(s) in ‘data’ (e.g. ‘plot_ly(mtcars, x
     = ~wt)’). Formulas are optional when supplying values directly,
     but they do help inform default axis/scale titles (e.g.,
     ‘plot_ly(x = mtcars$wt)’ vs ‘plot_ly(x = ~mtcars$wt)’)

_A_u_t_h_o_r(_s):

     Carson Sievert

_R_e_f_e_r_e_n_c_e_s:

     <https://plotly-r.com/overview.html>

_S_e_e _A_l_s_o:

        • For initializing a plotly-geo object: ‘plot_geo()’

        • For initializing a plotly-mapbox object: ‘plot_mapbox()’

        • For translating a ggplot2 object to a plotly object:
          ‘ggplotly()’

        • For modifying any plotly object: ‘layout()’, ‘add_trace()’,
          ‘style()’

        • For linked brushing: ‘highlight()’

        • For arranging multiple plots: ‘subplot()’,
          ‘crosstalk::bscols()’

        • For inspecting plotly objects: ‘plotly_json()’

        • For quick, accurate, and searchable plotly.js reference:
          ‘schema()’

_E_x_a_m_p_l_e_s:

     ## Not run:
     
     
     # plot_ly() tries to create a sensible plot based on the information you 
     # give it. If you don't provide a trace type, plot_ly() will infer one.
     plot_ly(economics, x = ~pop)
     plot_ly(economics, x = ~date, y = ~pop)
     # plot_ly() doesn't require data frame(s), which allows one to take 
     # advantage of trace type(s) designed specifically for numeric matrices
     plot_ly(z = ~volcano)
     plot_ly(z = ~volcano, type = "surface")
     
     # plotly has a functional interface: every plotly function takes a plotly
     # object as it's first input argument and returns a modified plotly object
     add_lines(plot_ly(economics, x = ~date, y = ~unemploy/pop))
     
     # To make code more readable, plotly imports the pipe operator from magrittr
     economics %>% plot_ly(x = ~date, y = ~unemploy/pop) %>% add_lines()
     
     # Attributes defined via plot_ly() set 'global' attributes that 
     # are carried onto subsequent traces, but those may be over-written
     plot_ly(economics, x = ~date, color = I("black")) %>%
      add_lines(y = ~uempmed) %>%
      add_lines(y = ~psavert, color = I("red"))
     
     # Attributes are documented in the figure reference -> https://plotly.com/r/reference
     # You might notice plot_ly() has named arguments that aren't in this figure
     # reference. These arguments make it easier to map abstract data values to
     # visual attributes.
     p <- plot_ly(palmerpenguins::penguins, x = ~bill_length_mm, y = ~body_mass_g)
     add_markers(p, color = ~bill_depth_mm, size = ~bill_depth_mm)
     add_markers(p, color = ~species)
     add_markers(p, color = ~species, colors = "Set1")
     add_markers(p, symbol = ~species)
     add_paths(p, linetype = ~species)
     ## End(Not run)

college_test <- (outer(gpa_test,iq_test,LEVEL=1,FUN=salary_model))
college_test

# a 
# 固定IQ和GPA的取值，然后考虑预测变量的取值对于响应变量的影响 
 
library(plotly)   #用于构建交互式的图表 
salary_model <- function(GPA,IQ,LEVEL){
  50+20*GPA+0.07*IQ+35*LEVEL+0.01*GPA*IQ-10*GPA*LEVEL
}
gpa_test <- seq(1,5,length=10)  # 人工构建从1到5，长度为10的等间隔的IQ值
iq_test <- seq(1,200,length=10) # 人工构建从1到200，长度为10的等间隔的GPA值

college_test <- t(outer(gpa_test,iq_test,LEVEL=1,FUN=salary_model))  #从构建的gpa以及iq值中，预测出的salary值，LEVEL=1表示大学学历，对获取的各种gpaxiq值的组会
high_school_test <- t(outer(gpa_test,iq_test,LEVEL=0,FUN=salary_model)) #同上，注意此处需要进行转置，否则获得的结果在x轴以及y轴上是反过来的！！！

plot_ly(x=gpa_test,y=iq_test) |>
  add_surface(   # 添加一个 3D 表面图
    x=gpa_test,y=iq_test,
    z=~college_test, # 指定 z 轴的数据为 College 学生的薪水
    colorscale=list(c(0, 1), c("rgb(173,216,230)", "rgb(0,0,139)")),  # 指定颜色渐变,浅蓝色到深蓝色
    colorbar=list(title="college") # 指定颜色条的标题
) |> 
add_surface(    # 添加另一个 3D 表面图
  x=gpa_test,y=iq_test,
  z=~high_school_test,
  colorscale=list(c(0, 1), c("rgb(255,182,193)", "rgb(139,0,0)")),  # 浅红色到深红色
  colorbar=list(title="high school")
) |>
layout(    # 设置图表布局      
  scene=list(  # 指定 3D 场景的轴标签
    xaxis = list(title = "GPA"),
    yaxis = list(title = "IQ"),
    zaxis = list(title = "Salary")  
  )  
)

# b
salary_model(GPA = 4.0,IQ = 110,LEVEL = 1)  # 预测大学学历的薪水，是137.1

# c
# 对于交叉项的效应，我们同样可以通过lm函数来进行计算
# 就像前面实践lab中的那样，summary(lm(medv ~ lstat * age, data = Boston)) ，我们需要检验这个交叉项的显著值，我们才能判定这个交叉项是否对于模型的预测有重要的影响
# 所以这个问题是false！

# 越灵活。参数越多的模型，其在训练集上的表现会越好，但是在测试集上的表现可能会越差，这就是过拟合的问题
# 所以立方项的模型拟合会比线性模型拟合具有更低的训练集的RSS，立方项的拟合只需要在高于一次项的项上的系数归0，就可以得到一个线性模型，总之就是更加灵活
# 所以立方项的模型拟合会比线性模型拟合具有更低的训练集的RSS

# b
# 测试集部分的RSS，实际上会呈现出U形曲线，所以无法确定

# c
# 还是立方项的拟合模型会导致更低的训练集的RSS，道理和前面一样，因为训练集上拟合的话，立方项的模型永远比线性模型拟合得更好

# d
# 无法回答，因为到底Y和X之间的非线性或者线性关系如何无法确定

# a
library(ISLR2)
pairs(Auto)

# b

names(Auto)
# 'mpg''cylinders''displacement''horsepower''weight''acceleration''year''origin''name'
Auto %>% select(-name)%>%cor() # 去掉最后一列非数值变量列

# c
lm.fit <- lm(mpg ~ . -name,data = Auto)
summary(lm.fit)

Call:
lm(formula = mpg ~ . - name, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.5903 -2.1565 -0.1169  1.8690 13.0604 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -17.218435   4.644294  -3.707  0.00024 ***
cylinders     -0.493376   0.323282  -1.526  0.12780    
displacement   0.019896   0.007515   2.647  0.00844 ** 
horsepower    -0.016951   0.013787  -1.230  0.21963    
weight        -0.006474   0.000652  -9.929  < 2e-16 ***
acceleration   0.080576   0.098845   0.815  0.41548    
year           0.750773   0.050973  14.729  < 2e-16 ***
origin         1.426141   0.278136   5.127 4.67e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.328 on 384 degrees of freedom
Multiple R-squared:  0.8215,	Adjusted R-squared:  0.8182 
F-statistic: 252.4 on 7 and 384 DF,  p-value: < 2.2e-16

# 从上面我们可以很清楚地看到响应变量和预测变量之间的关系，从p值以及系数的估计值的正负综合来判断

# 有统计学显著性的是displacement、weight、year 、origin

# year的系数是正的，表明mpg平均每年增加0.750773（单位）

# d
par(mfrow = c(2, 2))
plot(lm.fit)

# e

# 有7个预测变量，理论上来说，有C72 = 7!/(2! * 5!) = 21个两两相关的交叉项
# 加载必要的库
library(dplyr)

# 定义变量
variables <- c('cylinders', 'displacement', 'horsepower', 'weight', 'acceleration', 'year', 'origin')

# 生成所有两两变量组合的交叉项
combinations <- combn(variables, 2, simplify = FALSE)

# 循环遍历每个组合，拟合线性模型并输出摘要
for (combo in combinations) {
  interaction_term <- paste(combo, collapse = ":")
  formula <- as.formula(paste("mpg ~ . -name +", interaction_term))
  lm.fit <- lm(formula, data = Auto)
  cat("\n\n### Interaction Term:", interaction_term, "\n")
  print(summary(lm.fit))
}

Attaching package: ‘dplyr’


The following objects are masked from ‘package:stats’:

    filter, lag


The following objects are masked from ‘package:base’:

    intersect, setdiff, setequal, union


### Interaction Term: cylinders:displacement 

Call:
lm(formula = formula, data = Auto)

Residuals:
     Min       1Q   Median       3Q      Max 
-11.6081  -1.7833  -0.0465   1.6821  12.2617 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)            -2.7096590  4.6858582  -0.578 0.563426    
cylinders              -2.6962123  0.4094916  -6.584 1.51e-10 ***
displacement           -0.0774797  0.0141535  -5.474 7.96e-08 ***
horsepower             -0.0476026  0.0133736  -3.559 0.000418 ***
weight                 -0.0052339  0.0006253  -8.370 1.10e-15 ***
acceleration            0.0597997  0.0918038   0.651 0.515188    
year                    0.7594500  0.0473354  16.044  < 2e-16 ***
origin                  0.7087399  0.2736917   2.590 0.009976 ** 
cylinders:displacement  0.0136081  0.0017209   7.907 2.84e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.089 on 383 degrees of freedom
Multiple R-squared:  0.8465,	Adjusted R-squared:  0.8433 
F-statistic: 264.1 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: cylinders:horsepower 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.2399 -1.6871 -0.0511  1.2858 11.9380 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)          11.7025260  4.9115648   2.383 0.017676 *  
cylinders            -4.3060695  0.4580950  -9.400  < 2e-16 ***
displacement         -0.0013925  0.0069110  -0.201 0.840426    
horsepower           -0.3156601  0.0306339 -10.304  < 2e-16 ***
weight               -0.0038948  0.0006231  -6.250 1.09e-09 ***
acceleration         -0.1703028  0.0901427  -1.889 0.059612 .  
year                  0.7393193  0.0448736  16.476  < 2e-16 ***
origin                0.9031644  0.2496880   3.617 0.000338 ***
cylinders:horsepower  0.0402008  0.0037856  10.619  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.929 on 383 degrees of freedom
Multiple R-squared:  0.8621,	Adjusted R-squared:  0.8592 
F-statistic: 299.3 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: cylinders:weight 

Call:
lm(formula = formula, data = Auto)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.9484  -1.7133  -0.1809   1.4530  12.4137 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       7.3143478  5.0076737   1.461  0.14494    
cylinders        -5.0347425  0.5795767  -8.687  < 2e-16 ***
displacement      0.0156444  0.0068409   2.287  0.02275 *  
horsepower       -0.0314213  0.0126216  -2.489  0.01322 *  
weight           -0.0150329  0.0011125 -13.513  < 2e-16 ***
acceleration      0.1006438  0.0897944   1.121  0.26306    
year              0.7813453  0.0464139  16.834  < 2e-16 ***
origin            0.8030154  0.2617333   3.068  0.00231 ** 
cylinders:weight  0.0015058  0.0001657   9.088  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.022 on 383 degrees of freedom
Multiple R-squared:  0.8531,	Adjusted R-squared:  0.8501 
F-statistic: 278.1 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: cylinders:acceleration 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6499 -2.0696 -0.1522  1.7965 12.5285 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)            -3.424e+01  5.741e+00  -5.964 5.60e-09 ***
cylinders               2.940e+00  7.810e-01   3.764 0.000193 ***
displacement            8.611e-03  7.677e-03   1.122 0.262667    
horsepower             -3.805e-02  1.411e-02  -2.697 0.007308 ** 
weight                 -5.238e-03  6.844e-04  -7.653 1.61e-13 ***
acceleration            1.097e+00  2.324e-01   4.719 3.33e-06 ***
year                    7.654e-01  4.966e-02  15.412  < 2e-16 ***
origin                  1.269e+00  2.725e-01   4.657 4.44e-06 ***
cylinders:acceleration -2.141e-01  4.458e-02  -4.802 2.25e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.236 on 383 degrees of freedom
Multiple R-squared:  0.8316,	Adjusted R-squared:  0.8281 
F-statistic: 236.4 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: cylinders:year 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.3346 -2.0533 -0.1177  1.6257 12.0879 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)    -9.290e+01  1.256e+01  -7.399 8.75e-13 ***
cylinders       1.469e+01  2.378e+00   6.178 1.66e-09 ***
displacement    1.175e-02  7.259e-03   1.619   0.1063    
horsepower     -2.765e-02  1.322e-02  -2.092   0.0371 *  
weight         -6.242e-03  6.212e-04 -10.049  < 2e-16 ***
acceleration    1.213e-01  9.423e-02   1.287   0.1988    
year            1.734e+00  1.601e-01  10.826  < 2e-16 ***
origin          1.375e+00  2.647e-01   5.197 3.30e-07 ***
cylinders:year -1.949e-01  3.027e-02  -6.439 3.60e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.165 on 383 degrees of freedom
Multiple R-squared:  0.8389,	Adjusted R-squared:  0.8356 
F-statistic: 249.3 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: cylinders:origin 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.7908 -2.1679 -0.1234  1.9043 13.0306 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)      -1.792e+01  5.169e+00  -3.466 0.000587 ***
cylinders        -3.572e-01  5.459e-01  -0.654 0.513327    
displacement      1.919e-02  7.865e-03   2.439 0.015169 *  
horsepower       -1.656e-02  1.386e-02  -1.195 0.232926    
weight           -6.462e-03  6.541e-04  -9.879  < 2e-16 ***
acceleration      8.068e-02  9.896e-02   0.815 0.415449    
year              7.527e-01  5.141e-02  14.640  < 2e-16 ***
origin            1.838e+00  1.358e+00   1.353 0.176836    
cylinders:origin -9.985e-02  3.223e-01  -0.310 0.756898    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.332 on 383 degrees of freedom
Multiple R-squared:  0.8215,	Adjusted R-squared:  0.8178 
F-statistic: 220.4 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: displacement:horsepower 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.7010 -1.6009 -0.0967  1.4119 12.6734 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)             -1.894e+00  4.302e+00  -0.440  0.66007    
cylinders                6.466e-01  3.017e-01   2.143  0.03275 *  
displacement            -7.487e-02  1.092e-02  -6.859 2.80e-11 ***
horsepower              -1.975e-01  2.052e-02  -9.624  < 2e-16 ***
weight                  -3.147e-03  6.475e-04  -4.861 1.71e-06 ***
acceleration            -2.131e-01  9.062e-02  -2.351  0.01921 *  
year                     7.379e-01  4.463e-02  16.534  < 2e-16 ***
origin                   6.891e-01  2.527e-01   2.727  0.00668 ** 
displacement:horsepower  5.236e-04  4.813e-05  10.878  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.912 on 383 degrees of freedom
Multiple R-squared:  0.8636,	Adjusted R-squared:  0.8608 
F-statistic: 303.1 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: displacement:weight 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.9027 -1.8092 -0.0946  1.5549 12.1687 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)         -5.389e+00  4.301e+00  -1.253   0.2109    
cylinders            1.175e-01  2.943e-01   0.399   0.6899    
displacement        -6.837e-02  1.104e-02  -6.193 1.52e-09 ***
horsepower          -3.280e-02  1.238e-02  -2.649   0.0084 ** 
weight              -1.064e-02  7.136e-04 -14.915  < 2e-16 ***
acceleration         6.724e-02  8.805e-02   0.764   0.4455    
year                 7.852e-01  4.553e-02  17.246  < 2e-16 ***
origin               5.610e-01  2.622e-01   2.139   0.0331 *  
displacement:weight  2.269e-05  2.257e-06  10.054  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.964 on 383 degrees of freedom
Multiple R-squared:  0.8588,	Adjusted R-squared:  0.8558 
F-statistic: 291.1 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: displacement:acceleration 

Call:
lm(formula = formula, data = Auto)

Residuals:
   Min     1Q Median     3Q    Max 
-8.129 -1.899 -0.135  1.755 12.119 

Coefficients:
                            Estimate Std. Error t value Pr(>|t|)    
(Intercept)               -3.005e+01  4.737e+00  -6.343 6.36e-10 ***
cylinders                  2.136e-03  3.125e-01   0.007 0.994550    
displacement               7.022e-02  1.005e-02   6.989 1.24e-11 ***
horsepower                -5.515e-02  1.407e-02  -3.920 0.000105 ***
weight                    -4.211e-03  6.929e-04  -6.077 2.96e-09 ***
acceleration               7.530e-01  1.332e-01   5.653 3.09e-08 ***
year                       7.722e-01  4.811e-02  16.051  < 2e-16 ***
origin                     1.057e+00  2.671e-01   3.958 9.01e-05 ***
displacement:acceleration -4.855e-03  6.879e-04  -7.058 7.99e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.134 on 383 degrees of freedom
Multiple R-squared:  0.842,	Adjusted R-squared:  0.8387 
F-statistic: 255.2 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: displacement:year 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.9549 -1.9978 -0.0345  1.6232 12.2793 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -6.241e+01  7.811e+00  -7.990 1.60e-14 ***
cylinders          1.200e-02  3.133e-01   0.038   0.9695    
displacement       2.710e-01  3.663e-02   7.399 8.78e-13 ***
horsepower        -3.201e-02  1.318e-02  -2.429   0.0156 *  
weight            -5.860e-03  6.211e-04  -9.435  < 2e-16 ***
acceleration       1.059e-01  9.328e-02   1.136   0.2568    
year               1.334e+00  9.636e-02  13.848  < 2e-16 ***
origin             1.230e+00  2.638e-01   4.662 4.34e-06 ***
displacement:year -3.491e-03  4.997e-04  -6.988 1.25e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.138 on 383 degrees of freedom
Multiple R-squared:  0.8417,	Adjusted R-squared:  0.8384 
F-statistic: 254.5 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: displacement:origin 

Call:
lm(formula = formula, data = Auto)

Residuals:
     Min       1Q   Median       3Q      Max 
-11.2312  -2.0927  -0.1295   1.7287  12.2761 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)         -2.140e+01  4.827e+00  -4.434 1.21e-05 ***
cylinders           -4.731e-01  3.204e-01  -1.477 0.140577    
displacement         3.802e-02  9.768e-03   3.892 0.000117 ***
horsepower          -1.523e-02  1.367e-02  -1.114 0.265920    
weight              -5.940e-03  6.723e-04  -8.835  < 2e-16 ***
acceleration         4.578e-02  9.868e-02   0.464 0.642939    
year                 7.682e-01  5.087e-02  15.103  < 2e-16 ***
origin               3.810e+00  8.762e-01   4.349 1.76e-05 ***
displacement:origin -2.262e-02  7.891e-03  -2.866 0.004383 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.297 on 383 degrees of freedom
Multiple R-squared:  0.8252,	Adjusted R-squared:  0.8216 
F-statistic: 226.1 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: horsepower:weight 

Call:
lm(formula = formula, data = Auto)

Residuals:
   Min     1Q Median     3Q    Max 
-8.589 -1.617 -0.184  1.541 12.001 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        2.876e+00  4.511e+00   0.638 0.524147    
cylinders         -2.955e-02  2.881e-01  -0.103 0.918363    
displacement       5.950e-03  6.750e-03   0.881 0.378610    
horsepower        -2.313e-01  2.363e-02  -9.791  < 2e-16 ***
weight            -1.121e-02  7.285e-04 -15.393  < 2e-16 ***
acceleration      -9.019e-02  8.855e-02  -1.019 0.309081    
year               7.695e-01  4.494e-02  17.124  < 2e-16 ***
origin             8.344e-01  2.513e-01   3.320 0.000986 ***
horsepower:weight  5.529e-05  5.227e-06  10.577  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.931 on 383 degrees of freedom
Multiple R-squared:  0.8618,	Adjusted R-squared:  0.859 
F-statistic: 298.6 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: horsepower:acceleration 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.0329 -1.8177 -0.1183  1.7247 12.4870 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)             -32.499820   4.923380  -6.601 1.36e-10 ***
cylinders                 0.083489   0.316913   0.263 0.792350    
displacement             -0.007649   0.008161  -0.937 0.349244    
horsepower                0.127188   0.024746   5.140 4.40e-07 ***
weight                   -0.003976   0.000716  -5.552 5.27e-08 ***
acceleration              0.983282   0.161513   6.088 2.78e-09 ***
year                      0.755919   0.048179  15.690  < 2e-16 ***
origin                    1.035733   0.268962   3.851 0.000138 ***
horsepower:acceleration  -0.012139   0.001772  -6.851 2.93e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.145 on 383 degrees of freedom
Multiple R-squared:  0.841,	Adjusted R-squared:  0.8376 
F-statistic: 253.2 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: horsepower:year 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6314 -1.8896 -0.0724  1.5781 11.6140 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)     -9.723e+01  1.008e+01  -9.648  < 2e-16 ***
cylinders        8.146e-02  3.027e-01   0.269    0.788    
displacement     5.183e-03  7.072e-03   0.733    0.464    
horsepower       8.070e-01  9.496e-02   8.498 4.33e-16 ***
weight          -5.288e-03  6.112e-04  -8.651  < 2e-16 ***
acceleration    -2.894e-02  9.121e-02  -0.317    0.751    
year             1.830e+00  1.318e-01  13.887  < 2e-16 ***
origin           1.178e+00  2.558e-01   4.604 5.66e-06 ***
horsepower:year -1.140e-02  1.302e-03  -8.754  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.042 on 383 degrees of freedom
Multiple R-squared:  0.8512,	Adjusted R-squared:  0.8481 
F-statistic:   274 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: horsepower:origin 

Call:
lm(formula = formula, data = Auto)

Residuals:
   Min     1Q Median     3Q    Max 
-9.277 -1.875 -0.225  1.570 12.080 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)       -2.196e+01  4.396e+00  -4.996 8.94e-07 ***
cylinders         -5.275e-01  3.028e-01  -1.742   0.0823 .  
displacement      -1.486e-03  7.607e-03  -0.195   0.8452    
horsepower         8.173e-02  1.856e-02   4.404 1.38e-05 ***
weight            -4.710e-03  6.555e-04  -7.186 3.52e-12 ***
acceleration      -1.124e-01  9.617e-02  -1.168   0.2434    
year               7.327e-01  4.780e-02  15.328  < 2e-16 ***
origin             7.695e+00  8.858e-01   8.687  < 2e-16 ***
horsepower:origin -7.955e-02  1.074e-02  -7.405 8.44e-13 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.116 on 383 degrees of freedom
Multiple R-squared:  0.8438,	Adjusted R-squared:  0.8406 
F-statistic: 258.7 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: weight:acceleration 

Call:
lm(formula = formula, data = Auto)

Residuals:
   Min     1Q Median     3Q    Max 
-8.247 -2.048 -0.045  1.619 12.193 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)         -4.364e+01  5.811e+00  -7.511 4.18e-13 ***
cylinders           -2.141e-01  3.078e-01  -0.696 0.487117    
displacement         3.138e-03  7.495e-03   0.419 0.675622    
horsepower          -4.141e-02  1.348e-02  -3.071 0.002287 ** 
weight               4.027e-03  1.636e-03   2.462 0.014268 *  
acceleration         1.629e+00  2.422e-01   6.726 6.36e-11 ***
year                 7.821e-01  4.833e-02  16.184  < 2e-16 ***
origin               1.033e+00  2.686e-01   3.846 0.000141 ***
weight:acceleration -5.826e-04  8.408e-05  -6.928 1.81e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.141 on 383 degrees of freedom
Multiple R-squared:  0.8414,	Adjusted R-squared:  0.838 
F-statistic: 253.9 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: weight:year 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.9995 -1.8495 -0.1559  1.6061 11.7042 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -1.186e+02  1.338e+01  -8.864  < 2e-16 ***
cylinders    -1.218e-01  3.032e-01  -0.402   0.6881    
displacement  1.293e-02  7.019e-03   1.842   0.0663 .  
horsepower   -2.877e-02  1.286e-02  -2.236   0.0259 *  
weight        3.044e-02  4.652e-03   6.543 1.94e-10 ***
acceleration  1.447e-01  9.196e-02   1.574   0.1164    
year          2.084e+00  1.732e-01  12.033  < 2e-16 ***
origin        1.174e+00  2.597e-01   4.519 8.30e-06 ***
weight:year  -4.879e-04  6.097e-05  -8.002 1.47e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.084 on 383 degrees of freedom
Multiple R-squared:  0.847,	Adjusted R-squared:  0.8439 
F-statistic: 265.1 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: weight:origin 

Call:
lm(formula = formula, data = Auto)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.6056  -2.0253  -0.1315   1.6295  12.5217 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)   -2.482e+01  4.891e+00  -5.075 6.06e-07 ***
cylinders     -5.799e-01  3.171e-01  -1.829   0.0682 .  
displacement   8.833e-03  7.810e-03   1.131   0.2588    
horsepower    -1.068e-02  1.358e-02  -0.787   0.4321    
weight        -2.782e-03  1.084e-03  -2.566   0.0107 *  
acceleration   3.766e-02  9.729e-02   0.387   0.6989    
year           7.632e-01  4.998e-02  15.270  < 2e-16 ***
origin         6.633e+00  1.265e+00   5.244 2.60e-07 ***
weight:origin -2.309e-03  5.477e-04  -4.215 3.11e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.257 on 383 degrees of freedom
Multiple R-squared:  0.8294,	Adjusted R-squared:  0.8258 
F-statistic: 232.7 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: acceleration:year 

Call:
lm(formula = formula, data = Auto)

Residuals:
     Min       1Q   Median       3Q      Max 
-10.2277  -2.1428  -0.0839   1.8216  12.2997 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)       97.6199866 20.8291407   4.687 3.86e-06 ***
cylinders         -0.2163880  0.3148719  -0.687  0.49236    
displacement       0.0087776  0.0074936   1.171  0.24219    
horsepower        -0.0269212  0.0133812  -2.012  0.04493 *  
weight            -0.0058738  0.0006363  -9.232  < 2e-16 ***
acceleration      -7.1818103  1.2900806  -5.567 4.88e-08 ***
year              -0.7461986  0.2696927  -2.767  0.00593 ** 
origin             1.2630333  0.2691459   4.693 3.76e-06 ***
acceleration:year  0.0945502  0.0167501   5.645 3.22e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.202 on 383 degrees of freedom
Multiple R-squared:  0.8352,	Adjusted R-squared:  0.8317 
F-statistic: 242.6 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: acceleration:origin 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.4106 -1.8805 -0.2471  1.7891 11.9680 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)         -0.3327273  5.0570077  -0.066   0.9476    
cylinders           -0.5881258  0.3063127  -1.920   0.0556 .  
displacement         0.0086251  0.0073062   1.181   0.2385    
horsepower          -0.0250843  0.0131049  -1.914   0.0564 .  
weight              -0.0052351  0.0006439  -8.131 5.98e-15 ***
acceleration        -1.0340600  0.1896960  -5.451 8.98e-08 ***
year                 0.7623813  0.0482774  15.792  < 2e-16 ***
origin              -9.3089774  1.6109675  -5.779 1.56e-08 ***
acceleration:origin  0.6546959  0.0969263   6.755 5.34e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.15 on 383 degrees of freedom
Multiple R-squared:  0.8405,	Adjusted R-squared:  0.8371 
F-statistic: 252.2 on 8 and 383 DF,  p-value: < 2.2e-16


### Interaction Term: year:origin 

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6072 -2.0439 -0.0596  1.7121 12.3368 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)   8.492e+00  9.044e+00   0.939 0.348353    
cylinders    -5.042e-01  3.192e-01  -1.579 0.115082    
displacement  1.567e-02  7.530e-03   2.081 0.038060 *  
horsepower   -1.399e-02  1.364e-02  -1.025 0.305786    
weight       -6.352e-03  6.449e-04  -9.851  < 2e-16 ***
acceleration  9.185e-02  9.766e-02   0.941 0.347546    
year          4.189e-01  1.125e-01   3.723 0.000226 ***
origin       -1.405e+01  4.699e+00  -2.989 0.002978 ** 
year:origin   1.989e-01  6.030e-02   3.298 0.001064 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.286 on 383 degrees of freedom
Multiple R-squared:  0.8264,	Adjusted R-squared:  0.8228 
F-statistic: 227.9 on 8 and 383 DF,  p-value: < 2.2e-16

# 下面这种方法弃用！！！

# 定义变量
variables <- c('cylinders', 'displacement', 'horsepower', 'weight', 'acceleration', 'year', 'origin')

# 生成所有两两变量组合的交叉项
combinations <- combn(variables, 2, simplify = FALSE)

# 创建包含所有交叉项的公式
interaction_terms <- sapply(combinations, function(combo) paste(combo, collapse = ":"))
formula <- as.formula(paste("mpg ~ . -name +", paste(interaction_terms, collapse = " + ")))

# 拟合线性模型并输出摘要
lm.fit <- lm(formula, data = Auto)
summary(lm.fit)

Call:
lm(formula = formula, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-7.6303 -1.4481  0.0596  1.2739 11.1386 

Coefficients:
                            Estimate Std. Error t value Pr(>|t|)   
(Intercept)                3.548e+01  5.314e+01   0.668  0.50475   
cylinders                  6.989e+00  8.248e+00   0.847  0.39738   
displacement              -4.785e-01  1.894e-01  -2.527  0.01192 * 
horsepower                 5.034e-01  3.470e-01   1.451  0.14769   
weight                     4.133e-03  1.759e-02   0.235  0.81442   
acceleration              -5.859e+00  2.174e+00  -2.696  0.00735 **
year                       6.974e-01  6.097e-01   1.144  0.25340   
origin                    -2.090e+01  7.097e+00  -2.944  0.00345 **
cylinders:displacement    -3.383e-03  6.455e-03  -0.524  0.60051   
cylinders:horsepower       1.161e-02  2.420e-02   0.480  0.63157   
cylinders:weight           3.575e-04  8.955e-04   0.399  0.69000   
cylinders:acceleration     2.779e-01  1.664e-01   1.670  0.09584 . 
cylinders:year            -1.741e-01  9.714e-02  -1.793  0.07389 . 
cylinders:origin           4.022e-01  4.926e-01   0.816  0.41482   
displacement:horsepower   -8.491e-05  2.885e-04  -0.294  0.76867   
displacement:weight        2.472e-05  1.470e-05   1.682  0.09342 . 
displacement:acceleration -3.479e-03  3.342e-03  -1.041  0.29853   
displacement:year          5.934e-03  2.391e-03   2.482  0.01352 * 
displacement:origin        2.398e-02  1.947e-02   1.232  0.21875   
horsepower:weight         -1.968e-05  2.924e-05  -0.673  0.50124   
horsepower:acceleration   -7.213e-03  3.719e-03  -1.939  0.05325 . 
horsepower:year           -5.838e-03  3.938e-03  -1.482  0.13916   
horsepower:origin          2.233e-03  2.930e-02   0.076  0.93931   
weight:acceleration        2.346e-04  2.289e-04   1.025  0.30596   
weight:year               -2.245e-04  2.127e-04  -1.056  0.29182   
weight:origin             -5.789e-04  1.591e-03  -0.364  0.71623   
acceleration:year          5.562e-02  2.558e-02   2.174  0.03033 * 
acceleration:origin        4.583e-01  1.567e-01   2.926  0.00365 **
year:origin                1.393e-01  7.399e-02   1.882  0.06062 . 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.695 on 363 degrees of freedom
Multiple R-squared:  0.8893,	Adjusted R-squared:  0.8808 
F-statistic: 104.2 on 28 and 363 DF,  p-value: < 2.2e-16

# f

# 此处我们以horsepower为例，进行非线性变换拟合

par(mfrow = c(2, 2))
plot(Auto$horsepower, Auto$mpg)
plot(log(Auto$horsepower), Auto$mpg)
plot(sqrt(Auto$horsepower), Auto$mpg)
plot(Auto$horsepower^2, Auto$mpg)

par(mfrow = c(2, 2))
plot(Auto$horsepower, Auto$mpg, cex = 0.2)
plot(log(Auto$horsepower), Auto$mpg, cex = 0.2)
plot(sqrt(Auto$horsepower), Auto$mpg, cex = 0.2)
plot(Auto$horsepower^2, Auto$mpg, cex = 0.2)

# 此处我们仅以log转换为例

Auto %>% mutate(horsepower=log(horsepower))
fit.new <- lm(mpg ~ . -name,data = Auto)
summary(fit.new)
par(mfrow = c(2, 2))
plot(fit.new)

Auto <- Auto %>% mutate(horsepower=log(horsepower))
fit.new <- lm(mpg ~ . -name,data = Auto)
summary(fit.new)
par(mfrow = c(2, 2))
plot(fit.new)

Call:
lm(formula = mpg ~ . - name, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.5903 -2.1565 -0.1169  1.8690 13.0604 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -17.218435   4.644294  -3.707  0.00024 ***
cylinders     -0.493376   0.323282  -1.526  0.12780    
displacement   0.019896   0.007515   2.647  0.00844 ** 
horsepower    -0.016951   0.013787  -1.230  0.21963    
weight        -0.006474   0.000652  -9.929  < 2e-16 ***
acceleration   0.080576   0.098845   0.815  0.41548    
year           0.750773   0.050973  14.729  < 2e-16 ***
origin         1.426141   0.278136   5.127 4.67e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.328 on 384 degrees of freedom
Multiple R-squared:  0.8215,	Adjusted R-squared:  0.8182 
F-statistic: 252.4 on 7 and 384 DF,  p-value: < 2.2e-16

Call:
lm(formula = mpg ~ . - name, data = Auto)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.3115 -2.0041 -0.1726  1.8393 12.6579 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  27.254005   8.589614   3.173  0.00163 ** 
cylinders    -0.486206   0.306692  -1.585  0.11372    
displacement  0.019456   0.006876   2.830  0.00491 ** 
horsepower   -9.506436   1.539619  -6.175 1.69e-09 ***
weight       -0.004266   0.000694  -6.148 1.97e-09 ***
acceleration -0.292088   0.103804  -2.814  0.00515 ** 
year          0.705329   0.048456  14.556  < 2e-16 ***
origin        1.482435   0.259347   5.716 2.19e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.18 on 384 degrees of freedom
Multiple R-squared:  0.837,	Adjusted R-squared:  0.834 
F-statistic: 281.6 on 7 and 384 DF,  p-value: < 2.2e-16

# 可以发现在log转变之后horsepower似乎与mpg更呈现线性关系了，并且p值更加显著了

set.seed(1)

# a
X <- rnorm(n = 100,mean = 0,sd = 1)
print(X)

  [1] -0.626453811  0.183643324 -0.835628612  1.595280802  0.329507772
  [6] -0.820468384  0.487429052  0.738324705  0.575781352 -0.305388387
 [11]  1.511781168  0.389843236 -0.621240581 -2.214699887  1.124930918
 [16] -0.044933609 -0.016190263  0.943836211  0.821221195  0.593901321
 [21]  0.918977372  0.782136301  0.074564983 -1.989351696  0.619825748
 [26] -0.056128740 -0.155795507 -1.470752384 -0.478150055  0.417941560
 [31]  1.358679552 -0.102787727  0.387671612 -0.053805041 -1.377059557
 [36] -0.414994563 -0.394289954 -0.059313397  1.100025372  0.763175748
 [41] -0.164523596 -0.253361680  0.696963375  0.556663199 -0.688755695
 [46] -0.707495157  0.364581962  0.768532925 -0.112346212  0.881107726
 [51]  0.398105880 -0.612026393  0.341119691 -1.129363096  1.433023702
 [56]  1.980399899 -0.367221476 -1.044134626  0.569719627 -0.135054604
 [61]  2.401617761 -0.039240003  0.689739362  0.028002159 -0.743273209
 [66]  0.188792300 -1.804958629  1.465554862  0.153253338  2.172611670
 [71]  0.475509529 -0.709946431  0.610726353 -0.934097632 -1.253633400
 [76]  0.291446236 -0.443291873  0.001105352  0.074341324 -0.589520946
 [81] -0.568668733 -0.135178615  1.178086997 -1.523566800  0.593946188
 [86]  0.332950371  1.063099837 -0.304183924  0.370018810  0.267098791
 [91] -0.542520031  1.207867806  1.160402616  0.700213650  1.586833455
 [96]  0.558486426 -1.276592208 -0.573265414 -1.224612615 -0.473400636

# b
eps <- rnorm(n = 100,mean = 0,sd = sqrt(0.25)) # 注意这里是sd = sqrt(0.25)而不是0.25
print(eps)

  [1]  0.204700920  0.844436643  0.793294217 -0.165453900 -1.142617768
  [6]  1.248830795  0.333533083  0.270663668 -0.006699762  0.255054211
 [11] -0.082187916  0.210347322 -0.200123372 -0.685103939  0.493919134
 [16]  0.759872513 -0.154370285 -0.626644878  0.321120653 -0.022354568
 [21] -0.866609203  0.001065930 -0.315150167 -0.170484290 -0.578286181
 [26]  0.901570954 -0.165566018 -0.802756706  0.098596719  0.131587823
 [31] -0.492913350 -1.444460336 -0.320240851  0.285253818 -0.029861638
 [36] -0.049089372  0.280410364 -0.593229319  0.548388522 -0.002672014
 [41]  0.353655334  0.517053867  0.111740207 -0.439353806  0.581482278
 [46] -1.000082472 -0.272395370 -0.127835355 -0.083060518  0.510231954
 [51]  0.068110947  0.203583802 -0.034827407 -0.123832171  0.347775403
 [56]  0.573114179 -1.201548107  0.286369778  0.187362203 -0.212633861
 [61]  0.475506404 -0.194618591 -0.142165331  0.428704889  0.859813650
 [66]  0.135027450 -0.211092005 -0.594556647 -0.165516489 -0.469914663
 [71] -0.129466292  0.197189584 -0.425928546  1.324583441  0.078005838
 [76]  0.565103634 -1.144561990  0.370500579 -0.658122580  0.459901839
 [81]  0.199065078 -0.203764290  0.662129315 -0.350615835 -0.290307152
 [86] -0.500536091 -0.334089303  0.472592477  0.216851075  0.502579609
 [91] -0.195059332  0.188185146  0.122082462 -0.713128671  0.889214644
 [96]  0.067223830  0.382799500  0.477568338 -0.025282851 -0.152907710

# c
Y <- -1 + 0.5 * X + eps
print(Y)
length(Y)  # 向量长100

  [1] -1.10852599 -0.06374169 -0.62452009 -0.36781350 -1.97786388 -0.16140340
  [7] -0.42275239 -0.36017398 -0.71880909 -0.89763998 -0.32629733 -0.59473106
 [13] -1.51074366 -2.79245388  0.05638459 -0.26259429 -1.16246542 -1.15472677
 [19] -0.26826875 -0.72540391 -1.40712052 -0.60786592 -1.27786768 -2.16516014
 [25] -1.26837331 -0.12649342 -1.24346377 -2.53813290 -1.14047831 -0.65944140
 [31] -0.81357357 -2.49585420 -1.12640505 -0.74164870 -1.71839142 -1.25658665
 [37] -0.91673461 -1.62288602  0.09840121 -0.62108414 -0.72860646 -0.60962697
 [43] -0.53977810 -1.16102221 -0.76289557 -2.35383005 -1.09010439 -0.74356889
 [49] -1.13923362 -0.04921418 -0.73283611 -1.10242939 -0.86426756 -1.68851372
 [55]  0.06428725  0.56331413 -2.38515885 -1.23569754 -0.52777798 -1.28016116
 [61]  0.67631528 -1.21423859 -0.79729565 -0.55729403 -0.51182295 -0.77057640
 [67] -2.11357132 -0.86177922 -1.08888982 -0.38360883 -0.89171153 -1.15778363
 [73] -1.12056537 -0.14246538 -1.54881086 -0.28917325 -2.36620793 -0.62894675
 [79] -1.62095192 -0.83485863 -1.08526929 -1.27135360  0.25117281 -2.11239923
 [85] -0.99333406 -1.33406090 -0.80253938 -0.67949949 -0.59813952 -0.36387100
 [91] -1.46631935 -0.20788095 -0.29771623 -1.36302185  0.68263137 -0.65353296
 [97] -1.25549660 -0.80906437 -1.63758916 -1.38960803

# 我们开始拟合模型
lm.fit <- lm(Y ~ X)
summary(lm.fit)

# (Intercept) -0.98632，对应的就是截距的估计值，X            0.51058，对应的就是斜率的估计值

Call:
lm(formula = Y ~ X)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.45706 -0.24115 -0.02266  0.32462  1.32079 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.98632    0.05235 -18.840  < 2e-16 ***
X            0.51058    0.05815   8.781 5.34e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5197 on 98 degrees of freedom
Multiple R-squared:  0.4403,	Adjusted R-squared:  0.4346 
F-statistic:  77.1 on 1 and 98 DF,  p-value: 5.336e-14

# d

plot(X,Y)

# 从图上看起来，似乎是线性的

# e

# 我们开始拟合模型
lm.fit <- lm(Y ~ X)
summary(lm.fit)

# 看sd的话，两个系数拟合的挺好的
# 模型的系数估计值都非常显著，标准误较小，说明模型拟合良好。
# 截距项和 X 的系数都显著不为零，说明 X 对 y 有显著的线性影响。

Call:
lm(formula = Y ~ X)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.45706 -0.24115 -0.02266  0.32462  1.32079 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.98632    0.05235 -18.840  < 2e-16 ***
X            0.51058    0.05815   8.781 5.34e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5197 on 98 degrees of freedom
Multiple R-squared:  0.4403,	Adjusted R-squared:  0.4346 
F-statistic:  77.1 on 1 and 98 DF,  p-value: 5.336e-14

# f
plot(X, Y)
abline(lm.fit,col="red")

abline(-0.98632, 0.51058, col = "red", lty = 2)

legend("topleft",
  c("model fit", "population regression"),
  col = c("black", "red"),
  lty = c(1, 2)
)

# g

lm.fit <- lm(Y ~ X)
summary(lm.fit)

fit.2 <- lm(Y ~ X + I(X^2))
summary(fit.2)

# 或者
fit.22 <- lm(Y ~ poly(X, 2,raw = TRUE))
summary(fit.22)

anova(fit.2,lm.fit)
anova(fit.22,lm.fit)

# F检验都不显著，说明模型的拟合效果并没有提升

Call:
lm(formula = Y ~ X)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.45706 -0.24115 -0.02266  0.32462  1.32079 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.98632    0.05235 -18.840  < 2e-16 ***
X            0.51058    0.05815   8.781 5.34e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5197 on 98 degrees of freedom
Multiple R-squared:  0.4403,	Adjusted R-squared:  0.4346 
F-statistic:  77.1 on 1 and 98 DF,  p-value: 5.336e-14

Call:
lm(formula = Y ~ X + I(X^2))

Residuals:
     Min       1Q   Median       3Q      Max 
-1.46371 -0.24761 -0.01792  0.32978  1.32271 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.979434   0.064138 -15.271  < 2e-16 ***
X            0.511912   0.058865   8.696 8.71e-14 ***
I(X^2)      -0.008668   0.046208  -0.188    0.852    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5223 on 97 degrees of freedom
Multiple R-squared:  0.4405,	Adjusted R-squared:  0.429 
F-statistic: 38.19 on 2 and 97 DF,  p-value: 5.856e-13

Call:
lm(formula = Y ~ poly(X, 2, raw = TRUE))

Residuals:
     Min       1Q   Median       3Q      Max 
-1.46371 -0.24761 -0.01792  0.32978  1.32271 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)             -0.979434   0.064138 -15.271  < 2e-16 ***
poly(X, 2, raw = TRUE)1  0.511912   0.058865   8.696 8.71e-14 ***
poly(X, 2, raw = TRUE)2 -0.008668   0.046208  -0.188    0.852    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5223 on 97 degrees of freedom
Multiple R-squared:  0.4405,	Adjusted R-squared:  0.429 
F-statistic: 38.19 on 2 and 97 DF,  p-value: 5.856e-13

# h

# 比如说我们拟合方差为0.05的误差的模型

x <- rnorm(100, 0, 1)
y <- -1 + 0.5 * x + rnorm(100, 0, sqrt(0.05))
fit2 <- lm(y ~ x)
summary(fit2)
plot(x, y)
abline(fit2)
abline(-1, 0.5, col = "red", lty = 2)
legend("topleft",
  c("model fit", "population regression"),
  col = c("black", "red"),
  lty = c(1, 2)
)

# 可以看出，数据变异性减小，R2变高了

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.6547 -0.1600 -0.0127  0.1452  0.8618 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -1.00908    0.02631  -38.35   <2e-16 ***
x            0.50630    0.02663   19.01   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2628 on 98 degrees of freedom
Multiple R-squared:  0.7867,	Adjusted R-squared:  0.7845 
F-statistic: 361.4 on 1 and 98 DF,  p-value: < 2.2e-16

# i

# 噪声越高，我们就拟合1

x <- rnorm(100, 0, 1)
y <- -1 + 0.5 * x + rnorm(100, 0, 1)
fit3 <- lm(y ~ x)
summary(fit3)
plot(x, y)
abline(fit3)
abline(-1, 0.5, col = "red", lty = 2)
legend("topleft",
  c("model fit", "population regression"),
  col = c("black", "red"),
  lty = c(1, 2)
)

# 可以看出，数据变异性增大，R2变低了

Call:
lm(formula = y ~ x)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.73419 -0.80105 -0.05787  0.74623  2.87841 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -1.1992     0.1087 -11.027  < 2e-16 ***
x             0.5086     0.1130   4.501 1.87e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.086 on 98 degrees of freedom
Multiple R-squared:  0.1713,	Adjusted R-squared:  0.1628 
F-statistic: 20.26 on 1 and 98 DF,  p-value: 1.866e-05

# j
confint(lm.fit)
confint(fit2)
confint(fit3)

# 误差越小，置信区间越小

	crim	zn	indus	chas	nox	rm	age	dis	rad	tax	ptratio	lstat	medv
	<dbl>	<dbl>	<dbl>	<int>	<dbl>	<dbl>	<dbl>	<dbl>	<int>	<dbl>	<dbl>	<dbl>	<dbl>
1	0.00632	18	2.31	0	0.538	6.575	65.2	4.0900	1	296	15.3	4.98	24.0
2	0.02731	0	7.07	0	0.469	6.421	78.9	4.9671	2	242	17.8	9.14	21.6
3	0.02729	0	7.07	0	0.469	7.185	61.1	4.9671	2	242	17.8	4.03	34.7
4	0.03237	0	2.18	0	0.458	6.998	45.8	6.0622	3	222	18.7	2.94	33.4
5	0.06905	0	2.18	0	0.458	7.147	54.2	6.0622	3	222	18.7	5.33	36.2
6	0.02985	0	2.18	0	0.458	6.430	58.7	6.0622	3	222	18.7	5.21	28.7

	2.5 %	97.5 %
(Intercept)	33.448457	35.6592247
lstat	-1.026148	-0.8739505

	fit	lwr	upr
1	29.80359	29.00741	30.59978
2	25.05335	24.47413	25.63256
3	20.30310	19.73159	20.87461

	fit	lwr	upr
1	29.80359	17.565675	42.04151
2	25.05335	12.827626	37.27907
3	20.30310	8.077742	32.52846

	Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>
1	504	19472.38	NA	NA	NA	NA
2	504	19472.38	0	0	NA	NA

95.08000	95.91789	96.75579	97.59368	98.43158	99.26947	100.1074	100.9453	101.7832	102.6211	103.4589	104.2968	105.1347	105.9726	106.8105	107.6484	108.4863	109.3242	110.1621	111.0000
99.52889	100.41333	101.29778	102.18222	103.06667	103.95111	104.8356	105.7200	106.6044	107.4889	108.3733	109.2578	110.1422	111.0267	111.9111	112.7956	113.6800	114.5644	115.4489	116.3333
103.97778	104.90877	105.83977	106.77076	107.70175	108.63275	109.5637	110.4947	111.4257	112.3567	113.2877	114.2187	115.1497	116.0807	117.0117	117.9427	118.8737	119.8047	120.7357	121.6667
108.42667	109.40421	110.38175	111.35930	112.33684	113.31439	114.2919	115.2695	116.2470	117.2246	118.2021	119.1796	120.1572	121.1347	122.1123	123.0898	124.0674	125.0449	126.0225	127.0000
112.87556	113.89965	114.92374	115.94784	116.97193	117.99602	119.0201	120.0442	121.0683	122.0924	123.1165	124.1406	125.1647	126.1888	127.2129	128.2370	129.2611	130.2851	131.3092	132.3333
117.32444	118.39509	119.46573	120.53637	121.60702	122.67766	123.7483	124.8189	125.8896	126.9602	128.0309	129.1015	130.1722	131.2428	132.3135	133.3841	134.4547	135.5254	136.5960	137.6667
121.77333	122.89053	124.00772	125.12491	126.24211	127.35930	128.4765	129.5937	130.7109	131.8281	132.9453	134.0625	135.1796	136.2968	137.4140	138.5312	139.6484	140.7656	141.8828	143.0000
126.22222	127.38596	128.54971	129.71345	130.87719	132.04094	133.2047	134.3684	135.5322	136.6959	137.8596	139.0234	140.1871	141.3509	142.5146	143.6784	144.8421	146.0058	147.1696	148.3333
130.67111	131.88140	133.09170	134.30199	135.51228	136.72257	137.9329	139.1432	140.3535	141.5637	142.7740	143.9843	145.1946	146.4049	147.6152	148.8255	150.0358	151.2461	152.4564	153.6667
135.12000	136.37684	137.63368	138.89053	140.14737	141.40421	142.6611	143.9179	145.1747	146.4316	147.6884	148.9453	150.2021	151.4589	152.7158	153.9726	155.2295	156.4863	157.7432	159.0000

95.08000	99.52889	103.9778	108.4267	112.8756	117.3244	121.7733	126.2222	130.6711	135.1200
95.91789	100.41333	104.9088	109.4042	113.8996	118.3951	122.8905	127.3860	131.8814	136.3768
96.75579	101.29778	105.8398	110.3818	114.9237	119.4657	124.0077	128.5497	133.0917	137.6337
97.59368	102.18222	106.7708	111.3593	115.9478	120.5364	125.1249	129.7135	134.3020	138.8905
98.43158	103.06667	107.7018	112.3368	116.9719	121.6070	126.2421	130.8772	135.5123	140.1474
99.26947	103.95111	108.6327	113.3144	117.9960	122.6777	127.3593	132.0409	136.7226	141.4042
100.10737	104.83556	109.5637	114.2919	119.0201	123.7483	128.4765	133.2047	137.9329	142.6611
100.94526	105.72000	110.4947	115.2695	120.0442	124.8189	129.5937	134.3684	139.1432	143.9179
101.78316	106.60444	111.4257	116.2470	121.0683	125.8896	130.7109	135.5322	140.3535	145.1747
102.62105	107.48889	112.3567	117.2246	122.0924	126.9602	131.8281	136.6959	141.5637	146.4316
103.45895	108.37333	113.2877	118.2021	123.1165	128.0309	132.9453	137.8596	142.7740	147.6884
104.29684	109.25778	114.2187	119.1796	124.1406	129.1015	134.0625	139.0234	143.9843	148.9453
105.13474	110.14222	115.1497	120.1572	125.1647	130.1722	135.1796	140.1871	145.1946	150.2021
105.97263	111.02667	116.0807	121.1347	126.1888	131.2428	136.2968	141.3509	146.4049	151.4589
106.81053	111.91111	117.0117	122.1123	127.2129	132.3135	137.4140	142.5146	147.6152	152.7158
107.64842	112.79556	117.9427	123.0898	128.2370	133.3841	138.5312	143.6784	148.8255	153.9726
108.48632	113.68000	118.8737	124.0674	129.2611	134.4547	139.6484	144.8421	150.0358	155.2295
109.32421	114.56444	119.8047	125.0449	130.2851	135.5254	140.7656	146.0058	151.2461	156.4863
110.16211	115.44889	120.7357	126.0225	131.3092	136.5960	141.8828	147.1696	152.4564	157.7432
111.00000	116.33333	121.6667	127.0000	132.3333	137.6667	143.0000	148.3333	153.6667	159.0000

95.08000	96.84889	98.61778	100.3867	102.1556	103.9244	105.6933	107.4622	109.2311	111.0000
99.52889	101.39605	103.26321	105.1304	106.9975	108.8647	110.7319	112.5990	114.4662	116.3333
103.97778	105.94321	107.90864	109.8741	111.8395	113.8049	115.7704	117.7358	119.7012	121.6667
108.42667	110.49037	112.55407	114.6178	116.6815	118.7452	120.8089	122.8726	124.9363	127.0000
112.87556	115.03753	117.19951	119.3615	121.5235	123.6854	125.8474	128.0094	130.1714	132.3333
117.32444	119.58469	121.84494	124.1052	126.3654	128.6257	130.8859	133.1462	135.4064	137.6667
121.77333	124.13185	126.49037	128.8489	131.2074	133.5659	135.9244	138.2830	140.6415	143.0000
126.22222	128.67901	131.13580	133.5926	136.0494	138.5062	140.9630	143.4198	145.8765	148.3333
130.67111	133.22617	135.78123	138.3363	140.8914	143.4464	146.0015	148.5565	151.1116	153.6667
135.12000	137.77333	140.42667	143.0800	145.7333	148.3867	151.0400	153.6933	156.3467	159.0000

	mpg	cylinders	displacement	horsepower	weight	acceleration	year	origin
mpg	1.0000000	-0.7776175	-0.8051269	-0.7784268	-0.8322442	0.4233285	0.5805410	0.5652088
cylinders	-0.7776175	1.0000000	0.9508233	0.8429834	0.8975273	-0.5046834	-0.3456474	-0.5689316
displacement	-0.8051269	0.9508233	1.0000000	0.8972570	0.9329944	-0.5438005	-0.3698552	-0.6145351
horsepower	-0.7784268	0.8429834	0.8972570	1.0000000	0.8645377	-0.6891955	-0.4163615	-0.4551715
weight	-0.8322442	0.8975273	0.9329944	0.8645377	1.0000000	-0.4168392	-0.3091199	-0.5850054
acceleration	0.4233285	-0.5046834	-0.5438005	-0.6891955	-0.4168392	1.0000000	0.2903161	0.2127458
year	0.5805410	-0.3456474	-0.3698552	-0.4163615	-0.3091199	0.2903161	1.0000000	0.1815277
origin	0.5652088	-0.5689316	-0.6145351	-0.4551715	-0.5850054	0.2127458	0.1815277	1.0000000

A data.frame: 6 × 11
	Sales	CompPrice	Income	Advertising	Population	Price	ShelveLoc	Age	Education	Urban	US
	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<fct>	<dbl>	<dbl>	<fct>	<fct>
1	9.50	138	73	11	276	120	Bad	42	17	Yes	Yes
2	11.22	111	48	16	260	83	Good	65	10	Yes	Yes
3	10.06	113	35	10	269	80	Medium	59	12	Yes	Yes
4	7.40	117	100	4	466	97	Medium	55	14	Yes	Yes
5	4.15	141	64	3	340	128	Bad	38	13	Yes	No
6	10.81	124	113	13	501	72	Bad	78	16	No	Yes

A matrix: 3 × 2 of type dbl
	Good	Medium
Bad	0	0
Good	1	0
Medium	0	1

A data.frame: 392 × 9
	mpg	cylinders	displacement	horsepower	weight	acceleration	year	origin	name
	<dbl>	<int>	<dbl>	<dbl>	<int>	<dbl>	<int>	<int>	<fct>
1	18	8	307	4.867534	3504	12.0	70	1	chevrolet chevelle malibu
2	15	8	350	5.105945	3693	11.5	70	1	buick skylark 320
3	18	8	318	5.010635	3436	11.0	70	1	plymouth satellite
4	16	8	304	5.010635	3433	12.0	70	1	amc rebel sst
5	17	8	302	4.941642	3449	10.5	70	1	ford torino
6	15	8	429	5.288267	4341	10.0	70	1	ford galaxie 500
7	14	8	454	5.393628	4354	9.0	70	1	chevrolet impala
8	14	8	440	5.370638	4312	8.5	70	1	plymouth fury iii
9	14	8	455	5.416100	4425	10.0	70	1	pontiac catalina
10	15	8	390	5.247024	3850	8.5	70	1	amc ambassador dpl
11	15	8	383	5.135798	3563	10.0	70	1	dodge challenger se
12	14	8	340	5.075174	3609	8.0	70	1	plymouth 'cuda 340
13	15	8	400	5.010635	3761	9.5	70	1	chevrolet monte carlo
14	14	8	455	5.416100	3086	10.0	70	1	buick estate wagon (sw)
15	24	4	113	4.553877	2372	15.0	70	3	toyota corona mark ii
16	22	6	198	4.553877	2833	15.5	70	1	plymouth duster
17	18	6	199	4.574711	2774	15.5	70	1	amc hornet
18	21	6	200	4.442651	2587	16.0	70	1	ford maverick
19	27	4	97	4.477337	2130	14.5	70	3	datsun pl510
20	26	4	97	3.828641	1835	20.5	70	2	volkswagen 1131 deluxe sedan
21	25	4	110	4.465908	2672	17.5	70	2	peugeot 504
22	24	4	107	4.499810	2430	14.5	70	2	audi 100 ls
23	25	4	104	4.553877	2375	17.5	70	2	saab 99e
24	26	4	121	4.727388	2234	12.5	70	2	bmw 2002
25	21	6	199	4.499810	2648	15.0	70	1	amc gremlin
26	10	8	360	5.370638	4615	14.0	70	1	ford f250
27	10	8	307	5.298317	4376	15.0	70	1	chevy c20
28	11	8	318	5.347108	4382	13.5	70	1	dodge d200
29	9	8	304	5.262690	4732	18.5	70	1	hi 1200d
30	27	4	97	4.477337	2130	14.5	71	3	datsun pl510
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
368	28	4	112	4.477337	2605	19.6	82	1	chevrolet cavalier
369	27	4	112	4.477337	2640	18.6	82	1	chevrolet cavalier wagon
370	34	4	112	4.477337	2395	18.0	82	1	chevrolet cavalier 2-door
371	31	4	112	4.442651	2575	16.2	82	1	pontiac j2000 se hatchback
372	29	4	135	4.430817	2525	16.0	82	1	dodge aries se
373	27	4	151	4.499810	2735	18.0	82	1	pontiac phoenix
374	24	4	140	4.521789	2865	16.4	82	1	ford fairmont futura
375	36	4	105	4.304065	1980	15.3	82	2	volkswagen rabbit l

A matrix: 2 × 2 of type dbl
	2.5 %	97.5 %
(Intercept)	-1.0902064	-0.8824249
X	0.3951885	0.6259784